Abstract: The Stokes system is a linearized system of the Navier-Stokes equations describing the motion of incompressible viscous fluids. It is believed that the nonstationary problem is very close to the heat equation. (In fact, if one considers the Stokes system in a whole space R^n, the problem is reduced to the heat equation.) The solution operator S(t) of the Stokes system is called the Stokes semigroup. It is well-known that S(t) is analytic in the L^p setting for a large class of domains including bounded and exterior domains with smooth boundaries provided that p is finite and larger than 1. This property is the same as the heat semigroup. Moreover, for the heat semigroup it is analytic even when p equals the infinity.
The corresponding (p=infinity) result for the Stokes semigroup S(t) has been open for more than thirty years even if the domain is bounded. Using a blowup-argument, we have now solved this long-standing problem for a large class of domains, including bounded and exterior domains. A key step is to derive a harmonic pressure gradient estimate by a velocity gradient. We give a sketch of the proof as well as a few possible applications to the Navier-Stokes equations. This is a joint work of my student Ken Abe and the main paper is going to appear in Acta Mathematica.